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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses nonterminating divisions. By the end of the module students should understand the meaning of a nonterminating division and be able to recognize a nonterminating number by its notation.

Section overview

  • Nonterminating Divisions
  • Denoting Nonterminating Quotients

Nonterminating divisions

Let's consider two divisions:

  1. 9 . 8 ÷ 3 . 5 size 12{9 "." "8 " div " 3" "." 5} {}
  2. 4 ÷ 3 size 12{4 div " 3"} {}

Terminating divisions

Previously, we have considered divisions like example 1 , which is an example of a terminating division. A terminating division is a division in which the quotient terminates after several divisions (the remainder is zero ).

Long division. 9.8 divided by 3.5. The decimal place needs to be moved to the right one space, making the problem 98 divided by 35. 35 goes into 98 twice, with a remainder of 28. Bring down a zero to find the quotient in decimal form. 35 goes into 280 exactly 8 times. The quotient is 2.8

Exact divisions

The quotient in this problem terminates in the tenths position. Terminating divi­sions are also called exact divisions .

Nonterminating division

The division in example 2 is an example of a nonterminating division. A non-terminating division is a division that, regardless of how far we carry it out, always has a remainder .

Long division. 4 divided by 3 equals 1.333, with a repeating unresolved remainder, leading to a division problem that never terminates.

Repeating decimal

We can see that the pattern in the brace is repeated endlessly. Such a decimal quotient is called a repeating decimal .

Denoting nonterminating quotients

We use three dots at the end of a number to indicate that a pattern repeats itself endlessly.

4 ÷ 3 = 1 . 333 size 12{"4 " div " 3 "=" 1" "." "333 " dotslow } {}

Another way, aside from using three dots, of denoting an endlessly repeating pattern is to write a bar ( ¯ ) above the repeating sequence of digits.

4 ÷ 3 = 1 . 3 ¯ size 12{"4 " div " 3 "=" 1" "." {overline {3}} } {}

The bar indicates the repeated pattern of 3.

Repeating patterns in a division can be discovered in two ways:

  1. As the division process progresses, should the remainder ever be the same as the dividend, it can be concluded that the division is nonterminating and that the pattern in the quotient repeats. This fact is illustrated in [link] of [link] .
  2. As the division process progresses, should the "product, difference" pattern ever repeat two consecutive times, it can be concluded that the division is nonter­minating and that the pattern in the quotient repeats. This fact is illustrated in [link] and 4 of [link] .

Sample set a

Carry out each division until the repeating pattern can be determined.

100 ÷ 27 size 12{"100 " div " 27"} {}

3.70370 27 100.00000 81            ̲ 19 0         18 9         ̲ 100     81     ̲ 190   189  
When the remainder is identical to the dividend, the division is nonterminating. This implies that the pattern in the quotient repeats.

100 ÷ 27 = 3 . 70370370 size 12{"100" div "27"=3 "." "70370370" dotslow } {} The repeating block is 703.

100 ÷ 27 = 3 . 703 ¯ size 12{"100" div "27"=3 "." {overline {"703"}} } {}

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1 ÷ 9 size 12{"1 " div " 9"} {}

Long division. 1 divided by 9 a nonterminating division problem with a repeating quotient of .111
We see that this “product, difference”pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats.

1 ÷ 9 = 0 . 111 size 12{1 div 9=0 "." "111" dotslow } {} The repeating block is 1.

1 ÷ 9 = 0 . 1 ¯ size 12{1 div 9=0 "." {overline {1}} } {}

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Divide 2 by 11 and round to 3 decimal places.

Since we wish to round the quotient to three decimal places, we'll carry out the division so that the quotient has four decimal places.

.1818 11 2.0000 1 1       ̲ 90     88     ̲ 20   11   ̲ 90

The number .1818 rounded to three decimal places is .182. Thus, correct to three decimal places,

2 ÷ 11 = 0 . 182 size 12{"2 " div " 11 "=" 0" "." "182"} {}

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Divide 1 by 6.

Long division. 1 divided by six equals .166
We see that this “product, difference” pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats at the 6.

1 ÷ 6 = 0 . 1 6 ¯ size 12{1 div 6=0 "." 1 {overline {6}} } {}

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Practice set a

Carry out the following divisions until the repeating pattern can be determined.

1 ÷ 3 size 12{1 div 3} {}

0 . 3 ¯ size 12{0 "." {overline {3}} } {}

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5 ÷ 6 size 12{5 div 6} {}

0 . 8 3 ¯ size 12{0 "." 8 {overline {3}} } {}

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11 ÷ 9 size 12{"11" div 9} {}

1 . 2 ¯ size 12{1 "." {overline {2}} } {}

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17 ÷ 9 size 12{"17" div 9} {}

1 . 8 ¯ size 12{1 "." {overline {8}} } {}

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Divide 7 by 6 and round to 2 decimal places.

1.17

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Divide 400 by 11 and round to 4 decimal places.

36.3636

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Exercises

For the following 20 problems, carry out each division until the repeating pattern is determined. If a repeating pattern is not apparent, round the quotient to three decimal places.

4 ÷ 9 size 12{4 div 9} {}

0 . 4 ¯ size 12{0 "." {overline {4}} } {}

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8 ÷ 11 size 12{8 div "11"} {}

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4 ÷ 25 size 12{4 div "25"} {}

0.16

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5 ÷ 6 size 12{5 div 6} {}

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1 ÷ 7 size 12{1 div 7} {}

0 . 142857 ¯ size 12{0 "." {overline {"142857"}} } {}

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3 ÷ 1 . 1 size 12{3 div 1 "." 1} {}

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20 ÷ 1 . 9 size 12{"20" div 1 "." 9} {}

10.526

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10 ÷ 2 . 7 size 12{"10" div 2 "." 7} {}

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1 . 11 ÷ 9 . 9 size 12{1 "." "11" div 9 "." 9} {}

0 . 1 12 ¯ size 12{0 "." 1 {overline {"12"}} } {}

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8 . 08 ÷ 3 . 1 size 12{8 "." "08" div 3 "." 1} {}

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51 ÷ 8 . 2 size 12{"51" div 8 "." 2} {}

6 . 21951 ¯ size 12{6 "." {overline {"21951"}} } {}

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0 . 213 ÷ 0 . 31 size 12{0 "." "213" div 0 "." "31"} {}

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0 . 009 ÷ 1 . 1 size 12{0 "." "009" div 1 "." 1} {}

0 . 00 81 ¯ size 12{0 "." "00" {overline {"81"}} } {}

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6 . 03 ÷ 1 . 9 size 12{6 "." "03" div 1 "." 9} {}

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0 . 518 ÷ 0 . 62 size 12{0 "." "518" div 0 "." "62"} {}

0.835

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1 . 55 ÷ 0 . 27 size 12{1 "." "55" div 0 "." "27"} {}

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0 . 333 ÷ 0 . 999 size 12{0 "." "333" div 0 "." "999"} {}

0 . 3 ¯ size 12{0 "." {overline {3}} } {}

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0 . 444 ÷ 0 . 999 size 12{0 "." "444" div 0 "." "999"} {}

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0 . 555 ÷ 0 . 27 size 12{0 "." "555" div 0 "." "27"} {}

2 . 0 5 ¯ size 12{2 "." 0 {overline {5}} } {}

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3 . 8 ÷ 0 . 99 size 12{3 "." 8 div 0 "." "99"} {}

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Calculator problems

For the following 10 problems, use a calculator to perform each division.

7 ÷ 9 size 12{7 div 9} {}

0 . 7 ¯ size 12{0 "." {overline {7}} } {}

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8 ÷ 11 size 12{8 div "11"} {}

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14 ÷ 27 size 12{"14" div "27"} {}

0 . 518 ¯ size 12{0 "." {overline {"518"}} } {}

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1 ÷ 44 size 12{1 div "44"} {}

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2 ÷ 44 size 12{2 div "44"} {}

0 . 0 45 ¯ size 12{0 "." 0 {overline {"45"}} } {}

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0 . 7 ÷ 0 . 9 size 12{0 "." 7 div 0 "." 9} {} (Compare this with [link] .)

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80 ÷ 110 size 12{"80" div "110"} {} (Compare this with [link] .)

0 . 72 ¯ size 12{0 "." {overline {"72"}} } {}

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0 . 0707 ÷ 0 . 7070 size 12{0 "." "0707" div 0 "." "7070"} {}

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0 . 1414 ÷ 0 . 2020 size 12{0 "." "1414" div 0 "." "2020"} {}

0.7

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1 ÷ 0 . 9999999 size 12{1 div 0 "." "9999999"} {}

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Exercise for review

( [link] ) In the number 411,105, how many ten thousands are there?

1

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( [link] ) Find the quotient, if it exists. 17 ÷ 0 size 12{"17" div 0} {} .

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( [link] ) Find the least common multiple of 45, 63, and 98.

4410

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( [link] ) Subtract 8.01629 from 9.00187 and round the result to three decimal places.

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( [link] ) Find the quotient. 104 . 06 ÷ 12 . 1 size 12{"104" "." "06" div "12" "." 1} {} .

8.6

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Questions & Answers

more explanation on GDP
Isaac Reply
it is a country total out put of goods and services divided by the total population of the country.I think it can also be derived from the country labour force,,because it mostly depend on the labour force and the level of technology .
Tantoh
labour force and technological progress leads to greater production increases the GDP
Ahmed
What is economics?
Bubu Reply
by this time
Emmanuel
It is a social science that analyses production,distribution and consumption of goods and services
Emmanuel
A social science that study human behavior in relationship with decision making
Jessica
What are the typical patterns of GDP for a high-income economy like the United States in the long run and the short run?
mwangala Reply
What are the limitation and significant of macroeconomic
Usman Reply
explain the significance of concerpt of opportunity cost in planning
Mwanaid Reply
what is meant by the price elasticity of demand?
Martine Reply
when price of a commodity increase it's demand contracts , and whe the price of a commodity decreases it's demand expands so the degree of change in demand in response to change in own price of the commodity is called PED . Ed = percentage change in quantity demanded / percentage change in price
shaswat
What are the limitations of macroeconomic and their segnificant
Usman Reply
Discuss the role of competition in stimulating economic growth?
Daniel Reply
competition stimulate economic growth because in such types of economy,they is no monopoly power every supplier will want to produce to meet customers choice which brings about quality production and attract invested and customers into such economy
Koka
competition creates Monopoly because of economy of scale. it's not antithesis but different side of same coin
toko
competition result in high economic growth since every firm will intend to provide quality services and products to meet customers needs and requirements unlike in Monopoly situation where a firm just provide what it want to resulting in large stock piles of unwanted products ,ie inefficiency, howev
Mark
microeconomics study part of the economy but macroeconomic study the whole economy
Olokun Reply
studying the whole economy, solving the problem of the economy and building up the economy
Olokun
micro means small while macro means large
Olokun
standard of living is the footsteps of an economy because it plays important role for country to have crucial view about their budget ,import and export
Olokun
it will be differ because economic agent will only take their views on some part of household
Olokun
can opportunity cost be zero
OBED Reply
how many types of transportation do we have
Jacob
yes. when a customer's purchasing power is high, he may have d ability to purchase all he needs, dt makes opportunity cost zero
George
please can give more explanation on this question
OBED
what are the factors production
PETER Reply
Labour capital entrepreneurs
Leta
Land,capital, labour,and the entrepreneur
Tantoh
I will like to know use of calculus in economics
JHUMA Reply
do they use it in economics?
Pranav
I want to know if I should take calculus or statistics and probability my senior year of highschool
Yahir
yes for example in monopolistic competitive market..... TR=TC* & THIS CALCULATED BY CHANGING( DERIVATIVE LAW) MR =MC ** WILL BE THE FORMULA THAT USE.
Leta
please in which topic in economic is the question coming from.
Tantoh
from PCF in economics
Leta
why is unitary proportional to responsiveness
Etim Reply
any tip for igcse economics exam?pls
Stacey Reply
well
The
What is a market
Divine Reply
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
what are the variables that affect demand
Divine
price of the related goods 2 price of the given commodity 3 income of the consumer 4 taste and preference 5 expectation in the future price
John
pls the taste and preference
Nas
explain briefly
Nas
a consumer taste and preference commodity changes for a time the man becomes
John
sorry sorry
John
is when the price of a commodity becomes high and can't afford example Samsung instead of iPhone
John
consumers who have high intense for goods will purchase the goods even if the price of that commodity increases because he or she preferred that commodity.people will be prefer iphone as its price increase
Yussif
as usual bad taste of preference is when a consumer regrets from one commodity to another in terms of the price
John
thanks alot
Nas
you're welcome
John
#Preference; #Income #Test
Dereje
#price Of Commodity #Income #Taste #Preference
Dereje
#Market is The Place Where Buyers And Sellers Are Exchanging Their Goods And service. #
Dereje
difference between macro and micro economics
Lawrence
Microeconomic Study about individual consumers market But Macroeconomis Study General economic Process Such As #Aggregate Demand #Aggregate Supples #GDp= #GNp
Dereje
nice so can u run down a brife discussion on GDP
Lawrence
good
Chinex
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
7hours 36 min - 4hours 50 min
Tanis Reply

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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